Longer gaps between values of binary quadratic forms

We prove new lower bounds on large gaps between integers that are sums of two squares or are represented by any binary quadratic form of discriminant D⁠, improving the results of Richards. Let s1,s2,… be the sequence of positive integers, arranged in increasing order, that are representable by any binary quadratic form of fixed discriminant D⁠, then lim supn→∞sn+1−snlogsn≫|D|φ(|D|)log|D|, improving a lower bound of 1|D| of Richards. In the special case of sums of two squares, we improve Richards’s bound of 1/4 to 390449=0.868…⁠. We also generalize Richards’s result in another direction: if d is composite we show that there exist constants Cd such that for all integer values of x none of the values pd(x)=Cd+xd is a sum of two squares. Let d be a prime. For all k∈N⁠, there exists a smallest positive integer yk such that none of the integers yk+jd,1≤j≤k⁠, is a sum of two squares. Moreover, lim supk→∞klogyk≫1logd−−−−√